From the empty set to higher infinity
Build the standard number systems from the empty set, then continue through ordinal and cardinal hierarchies, Choice, independence, and large-cardinal notions.
Story · 37 guided steps
Open storyStories & Views
Views apply curated filters, relation types, and display settings. Stories add a numbered concept sequence with Previous and Next navigation. You can adjust either while keeping its URL as long as its required nodes remain visible.
Featured stories and views
Experimental discovery
Follow the experiments, observations, and anomalies that forced physics to change—and the later tests that decided whether the new theories survived.
Story · 20 guided steps
Open storyFrom sets to spaces
Trace how bare collections acquire relations, distance, topology, measure, and analytic structure.
Story · 19 guided steps
Open storyAlgebraic structures
Explore how operations and laws organize groups, rings, modules, algebras, representations, and their categorical relationships.
Story · 18 guided steps
Open storyThe geometry of gravity
Connect differential geometry to spacetime, gravitation, Einstein’s equations, and their classical limits and tests.
Story · 19 guided steps
Open storyThe quantum revolution
Move from the empirical crises of classical physics through quantum mechanics, quantum fields, and precision tests.
Story · 18 guided steps
Open storyScale, renormalization, and effective theory
See why physical laws are organized by resolution scale, with running couplings, coarse-graining, and effective descriptions connecting high- and low-energy regimes.
Story · 21 guided steps
Open storyFrom symmetry to particle types
Follow the path from group actions to irreducible unitary representations, spinors, chirality, multiplets, and particle classifications.
Story · 22 guided steps
Open storyQuantum states, measurement, and entanglement
Trace the operational structure of quantum theory from states and observables through measurement, composite systems, Bell tests, and decoherence.
Story · 23 guided steps
Open storyGauge fields as geometry
Translate local gauge invariance into principal bundles, connections, curvature, Yang–Mills theory, and the gauge sectors of the Standard Model.
Story · 19 guided steps
Open storyLocal-to-global geometry: sheaves and schemes
See how local data, germs, localization, gluing, and affine spectra build schemes and their geometric refinements.
Story · 27 guided steps
Open storyThe neutrino changes identity
Follow the neutrino from the beta-spectrum problem through direct detection, flavor, helicity, solar and atmospheric anomalies, mixing, and nonzero mass.
Story · 17 guided steps
Open storyHomotopy, homology, and cohomology
Compare deformation-based and algebraic invariants of spaces, from loops and homotopy groups to chain complexes, homology, cohomology rings, and exact sequences.
Story · 22 guided steps
Open storyComputability and incompleteness
Follow the formalization of algorithms to universal machines, undecidability, arithmetized syntax, and Gödel’s limits on effective formal theories.
Story · 19 guided steps
Open storyBlack holes and cosmology
Extend general relativity from exact black-hole geometries and gravitational collapse to FLRW cosmology, the hot Big Bang, inflation, dark components, and 𝛬CDM.
Story · 20 guided steps
Open storyLocal–global principles in arithmetic
Compare global rational solutions with solutions over every real and 𝑝-adic completion, including both Hasse–Minkowski success and genuine failures.
Story · 17 guided steps
Open storyFrom atomic identity to chemical reactivity
Trace how proton number and electronic organization become periodic patterns, bonds, molecular structures, mechanisms, and catalysis.
Story · 14 guided steps
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Foundations and logic
Survey the concepts used to formalize mathematical objects, proof, models, and foundational systems.
Story · 18 guided steps
Open storyAnatomy of the Standard Model
Inspect the fields, particles, symmetries, interactions, and measurements assembled into the Standard Model.
Story · 21 guided steps
Open storyFrom classical to relativistic physics
See how mechanics and electromagnetism are reorganized by spacetime symmetry and recovered as controlled limits.
Story · 19 guided steps
Open storyProbability, measure, and analysis
Connect measurable spaces, integration, probability distributions, function spaces, and stochastic structure.
Story · 14 guided steps
Open storyFrom action principles to quantum fields
Connect variational mechanics, Hamiltonian and symplectic structure, quantization, Fock space, perturbation theory, and scattering.
Story · 25 guided steps
Open storyFrom quantum fields to ordinary matter
Follow the hierarchy from quantum fields and elementary excitations through hadrons, nuclei, atoms, and molecules.
Story · 23 guided steps
Open storyThe hierarchy of topological properties
Explore separation, countability, metric, compactness, local, connectedness, and zero-dimensionality properties as a branching implication lattice.
Story · 24 guided steps
Open storySpaces, operators, and spectra
Build Banach and Hilbert spaces, then develop bounded and unbounded operators, adjoints, spectra, spectral measures, and Fock-space constructions.
Story · 27 guided steps
Open storyFrom orders to logics and spaces
Develop preorders, lattices, Boolean and Heyting algebras, frames, locales, and Kripke semantics from binary relations.
Story · 18 guided steps
Open storyManifolds and geometric structure
Build smooth manifolds and then layer tangent, cotangent, bundle, differential-form, connection, curvature, metric, symplectic, complex, and contact structures.
Story · 22 guided steps
Open storyLocal arithmetic and the 𝑝-adic world
Pass from number fields and Dedekind domains to valuations, places, completions, local fields, valuation rings, residue fields, and 𝑝-adic fields.
Story · 19 guided steps
Open storyThe atom becomes calculable
Follow the historical and mathematical route from spectral regularities and nuclear atoms to orbitals, many-electron methods, precision structure, and transitions.
Story · 30 guided steps
Open storyThe structure of the atomic nucleus
Trace nuclear discovery, forces, many-body models, binding, shell structure, isotopes, and radioactive decay.
Story · 22 guided steps
Open storyArchitectures of category theory
Explore the basic language of functors, natural transformations, universal constructions, and the major structural families of categories.
Story · 27 guided steps
Open storyGalois theory: polynomial equations and symmetry
See how field extensions, splitting fields, automorphism groups, and subgroup lattices encode polynomial solvability.
Story · 15 guided steps
Open storyEquilibrium becomes measurable voltage
Connect chemical potential and Gibbs energy to activities, acid–base and solubility equilibria, electrodes, the Nernst relation, and working cells.
Story · 14 guided steps
Open storyEvidence that shaped chemistry
Follow experiments that motivated chemical concepts and later measurements that tested their predictions, scope, and limits.
Story · 14 guided steps
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